Abstract

In the theory of causal fermion systems, the physical equations are obtained as the Euler–Lagrange equations of a causal variational principle. Studying families of critical measures of causal variational principles, a class of conserved surface layer integrals is found and analyzed.

Mathematics Subject Classification

Notes

Acknowledgements

We would like to thank Andreas Platzer and the referee for helpful comments on the manuscript.

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Appendix A: Combinatorics of the perturbation expansion

We now give the proof of Lemma 3.6. In order to facilitate reading, we give the details of the combinatorics. We note that a more compact proof, where the perturbation series is rewritten with exponentials, is given in [7, Proof of Lemma 4.1].

Proof of Lemma 3.6

Our task is to expand the functional \(I_m^\Omega \) in Theorem 3.1 in powers of \(\lambda \). We first note that (3.22) implies that

where in the second step we used that \(c^{(0)} = 0\) in order to truncate the summation over l as well as the general Leibniz rule for differentiable functions.

We next perform the perturbation expansion of the Lagrangian. To this end, we first expand it in a Taylor series in both arguments. In the following formula, \((D_{1, F(x)-x}) {\mathcal {L}}(x,y)\) and \((D_{2, F(y)-y}) {\mathcal {L}}(x,y)\) again denote the partial derivatives of \({\mathcal {L}}(.,.)\) acting on the first and second argument, in direction of \(F(x)-x\) and \(F(y)-y\), respectively. According to our general convention (3.16), these partial derivatives do not act on the arguments \(F(x)-x\) or \(F(y)-y\) of other derivative, but merely on \({\mathcal {L}}\). We thus obtain

where the second step consists merely of a reordering of summation indices, and where in the last step we used the generalized binomial theorem. Taking the \(q^\text {th}\) derivative with respect to \(\lambda \) and evaluating at \(\lambda =0\), the general Leibniz rule yields

where in the last step we argued as follows. Note that for a given value of (a, k), in the second-to-last line, the sum over q gives \((p-(a+k))\) terms since \(\ell _i,\,q_i \ge 1\). The sum over these \((p-(a+k))\) terms reads

where in the second step we used that extending the range of q does not change the terms which are summed over, because the conditions \(\ell _i,\,q_i \ge 1\) exclude those terms for fixed k and a. It remains to sum over the different values of (a, k), which is most conveniently done via

The last step can be understood as follows. In order to get from the second line to the third line, from every bracket \((c^{(q_i)}(x)+D_{1,F^{(q_i)}})\) we choose either \(c^{(q_i)}(x)\) or \(D_{1,F^{(q_i)}}\). The order of the appearance of the derivatives does not matter because, as explained after (A.1), the derivatives all act on the first argument of \({\mathcal {L}}(x,y)\) and not on each other nor on the \(c^{(q_i)}(x)\). Furthermore, the actual value of \(q_i\) does not matter because we sum over all \(q_i\). Thus every term in the second line is characterized by how many \(c^{(q_i)}(x)\) appear (denote this number be a) and by how many \(D_{1,F^{(q_i)}}\) appear (we denote this number be k), giving rise to the sum over a and k in the last line. The binomial coefficient gives the correct combinatorial factor, corresponding to the number of ways that, disregarding ordering, we can choose a (respectively k) terms from l terms.

The same argument as in the last paragraph can be applied to terms \((\nabla _{1,\mathfrak {w}^{(q_i)}} + \nabla _{2,\mathfrak {w}^{(q_i)}})\) instead of \(\nabla _{1,\mathfrak {w}^{(q_i)}}\), giving exactly the terms appearing in the last term in (A.2). Thus the \(p^\text {th}\) order in the perturbation expansion of the integrand of (3.9) can be expressed as

The remaining task is to distribute the s- and t-derivatives in Theorem 3.1. Since the term for \(\ell =0\) in (A.3) only contributes if \(p=0\) and thus, according to (3.20) and (3.21), does not have a dependence on s or t, the summation over \(\ell \) does not need to include the case \(\ell =0\). We thus obtain

Note that the summation over \(k_1, \ldots , k_\ell \) needs to include the cases \(k_i = 0\) because in general \(\mathfrak {w}^{(p)}_{s,t} |_{s=0=t} \ne 0\) (see (3.23)). For the integrand in (3.10), using (A.1), we obtain for the \(p^\text {th}\) order